Darts

Darts is a fabulous sport. You can play it with your friends in local pubs or
you can invite the neighbours to a darts-evening. You will never get bored with
darts.

Darts is no longer just a pub sport. Millions of darts-fans all over the
world have already hung a dart-board on their wall. There is no other
play-ground where you can play so many different games as on the Clock-board.

In the following pages you can see some quotations of The Dart Book.

Kari Kaitanen 1996

The Dart Book

In this book you can find dozens of different interesting dart-games.
Numerous examples with clear pictures welcome all new players to this great
sport. You also learn how to throw more accurately.

For experts this book offers a deep and scientific study of darts. For the
first time all the finishes of 01-games have been optimised with a computer from
every point of the game five turns ahead. Just by knowing the absolute best
finish for your level you will greatly improve your chances of winning.

The new generation of dart-games is rapidly increasing in popularity in
Finland. The first real strategy game in darts, Mawari, is an
excellent choice from beginners to experts. As Snooker is called billiard's
chess, Mawari is the chess of darts.

Different levels of players

In every sport there are those who win championships. Then there are those
who are in it as a hobby. There are many different levels and many ways to throw
darts. There are those who make a living with it and those who just enjoy the
social life that darts provides - and usually spend their salary in places where
they meet. Both players - in my opinion - live a very full life and probably die
with a big smile on their face.

For the lesser players, as in every sport, it's almost impossible to beat a
master. This is also true in darts. However, it's typical of darts that the game
is interesting even if the players are of different levels. With a little luck a
beginner can beat an average player. But what's also typical of darts is that
there are not many sports where players of different levels can and should do so
much so differently from each other. The right choices depend on the level you
are playing at. This is if you want to win the game.

It's very useful to watch the decisions masters make and learn from them. But
when for example speaking about finishes in 501 (or 301), all the masters have
their own favourite finishes. Preferences speak more loudly when you try to find
the best 'out', and there is not one objective player in the world to say which
'out' is the best. So this is not the way you can find the best finishes for
your level. There is only one way to solve the problem. You must programme the
computer to throw with the different - but exact - sprays of darts.

At first we must learn the figure of the dispersion. There is one old
dart-player (there is no historical basis for this), Carl Friedrich Gauss
(1777-1855), who helped us in this enormous task. Without his discovery of
normally diversed phenomenon, it would be hard to calculate the probabilities
exactly. Gauss's dispersion proved to depict the most natural symptoms in life.
So we chose it to depict the distribution of dart shots.

Levels of players in The Dart Book

After solving how the darts are spread around the aiming point you must
decide which kind of players you want to take into your calculations. I decided
to use four different levels of players. Just to make it easier for you to know
what level I am talking about, I have named these levels: professional,
expert, average and beginner.

Since the dispersions of darts should be exact for finding the exact
percentages for these levels, there must be something that makes them exact. I
have chosen the probability of hitting the treble-bed. The 'professional's'
probability of hitting the treble is hereby always 46.42% in the book. An expert
can hit the treble with the probability of 21.54%. For an average player
the same figure is 10.00% and for the beginner only 4.64%.

I have not chosen these numbers randomly: the percentage of a professional
raised to the third power makes exactly 10%. You can thus say that a 'pro' can
get 180 points - three darts to treble-bed - every tenth time he tries. As you
may already have guessed an expert can score 180p every 100th time and so
on.

You must, however, notice that the probability of getting 180 points is only
theoretical: (46.4%^3 = 10%). After the first two darts there is already
considerably less space on the treble-bed for the third dart. Therefore you
often have to change your aiming point. Thus the 'right' probability of scoring
the maximum score may be one-to-twenty or even less.

Then there is the mental pressure, which may be enormous. If you have never
reached 180 points before, your hands may start to tremble. If you have
practiced darts for many years already and have never scored 180, it is probably
the biggest reason for this. It doesn't mean that you must now rank yourself in
the 'beginners' group. In fact, even if you try to test the level of your
play to find it out, there is no reason to start calling yourself 'an expert'
or 'an average player'. As mentioned earlier, the names are used only
because it's much easier to understand what kind of sprays I'm talking about.
However, the test can help you to see what level you should be interested in.

On the other hand, after the first dart the beginner e.g. might have a good 'sighter'
for his next shots, and the probability of throwing the remaining darts into the
target can be much bigger than before. So, it's very hard to tell the exact
level of scoring 180 points.

It would be more correct to say that an expert can hit three different
trebles, for example T20, T15 and T16, every hundredth time. Since the
theoretical probability is approximately right, it's easier to say that an
expert can get 180 points every tenth time.

The exact dispersions of darts

The next task is to iterate the dispersions to receive the exact required
percentage. Compared to the first book (published in Finnish) there are many
minor improvements. First, the probabilities are calculated much more
accurately; instead of using circles of four millimetres, the areas around the
aiming point are divided into parts smaller than 0.05 mm. The difference is
significant for example in the bull's-eye, the smallest target on the board.

The computer is programmed to imitate the real world as closely as possible.
For example, the wires hamper the players. The critical wire width is programmed
to be 0.2 mm. This means that an expert aiming at the centre of a treble-bed
loses 1.7% of his shots as 'wire-darts'. In the bull's-eye the probability of
wire-darts is already 3.1% for the expert.

Note that the wires are, of course, wider than the given 0.2 mm, but if the
dart hits a bit on the side of a wire it still goes in. It is not possible to
know the exact percentage of wire-darts, but it is important to tell the
computer that it's more probable in some areas - like in the bull's-eye - than
in the centre of singles. Nowadays technology makes darts more flexible and the
percentage of wire-darts is getting smaller.

Professional (pro)

When the term 'professional
player' (pro) is used in the book it means a player who can throw
(theoretically) 180 points every 10th time. The dispersion of darts is thus very
small. You can hear the great 'one-hundred-and-eightyyy...' shouted at the
biggest tournaments often enough to know that this level is not just
theoretical. However, there aren't many players in the world who represent this
honourable level of play.

As you can see in the picture, the area of possible hits is small, when a
'pro' aims at the centre of a treble-bed. The dispersion of this level is
programmed to the computer, and it uses exactly the same spray of darts when
aiming at any point on the board. When the dispersion is estimated to be
normally divided, you can calculate the exact probabilities for this level.

Expert player (exp)

The spray of darts is still
quite small. If an 'expert player' (exp) aims at the middle of triple-20, no
darts are missed over sectors 1 and 5. The possibility of throwing 180 points is
one in a hundred (1:100). The player of this level most certainly plays in a
darts-league. Since this level is already represented by a much greater amount
of players - and is still a very high level - I have tried to mention the
'expert' player often.

The spray of darts around treble-20 already looks more human and possible to
reach than the one of a 'professional player'. Already 16% of his darts fly to
the adjacent sectors. The average scoring is still very high: 74.7 points with
three darts (without doubling). This level can be the result of a great deal of
practice and years of social life in local pubs.

Average player (ave)

The level of an average
player can be very near the next one. The spray of hits is already quite large.
Only one tenth of the darts hit triple-20 when aiming at the middle of it.
Theoretically an average player (ave) is projected to throw 180 points every
1000th time. A few darts already miss over one sector and when for example
trying the heart of treble-20, one dart in every thirteenth turn hits sector 18
or sector 12.

Only one tenth of the shots at a treble are successful. Note that the aiming
point often has to be on the other side of a treble, when the bed is already
crowded. Move the same spray of darts only one centimetre to the right and
already every 26th dart (3.8%) hits sector 18 - meaning one dart every ninth
turn. Does this look more like your play?

Beginner (beg)

Beginners are the group of
players where levels vary the most. But when using the word 'beginner' (beg) in
the book, it means that the spray of darts draws quite a large pancake around
the aiming point. The 'beginner' finds it difficult to hit the same sector that
he aims at and thus the average score is not much different in any area of the
dart-board. 180 points is almost never achieved (1:10000).

A beginner finds it very frustrating to aim at a double-band. At first only
one shot in fourteen will score. And then there is the baby in the next room who
starts to cry every time you hit the enclosure cabinet. And these total misses
are not few at the beginning of your road to the championship tournaments. Just
a little practice and you will improve your play a great deal.

Theory

Note that in the 'real world' the dispersion of darts isn't usually exactly a
circle, but an ellipse. This due to the way one throws darts at the dart-board.
It's often more difficult to release the darts at the same point every time than
to keep the same vertical line. This means the spray is more often high than
wide. Of course there are players who do the opposite - they are the ones who
are more likely to hit the single of sector six (and double-top) than the
single-20.

To be more exact the ellipse may not be exactly vertical. For the
right-handed players the figure usually leans a bit to the left. This comes from
the motion of the hand when throwing. It also gives one more reason to use
double-16 (and double-8) in doubling out.

For master players, however, the dispersion is closer to a circle and for
them every area on the dart-board is quite as good.

Probabilities

As you have noticed, everything in darts has something to do with
probabilities. When you choose the way of finishing you just try to find the
best probability to win the game. The probability of the opponent finishing the
next turn is also important. You can estimate these probabilities in your head
or you can even calculate them mathematically, as you have seen done in the book
first time ever.

At first you must find out the dispersions of darts for the player. You can
do that with a test, but that would require a lot of throws to be reliable. Or
you can do it purely mathematically when you know the theory of normal
dispersion. Since we want to make the results of optimal play as general as
possible, we must use a circle as the form of the dispersion. One way or another
you have to know what the player's probability of hitting for example a double
is, how many darts then fly to a single, which percentage to the sector beside
and so on. These probabilities have to be known for every aiming point you want
to calculate.

When you have found a suitable spray of darts, you simply move it around the
dart-board. When you place exactly the same dispersion to the centre of a double
you get the percentage of hitting the double. Note that for the computer there
are three aiming points in every bed of treble, double and single.

Naturally, you must also use the same spray of darts for the bull's-eye.

Finding finishes with the computer

There are many variations for finishing 501. When placing only one
aiming point in the middle of each bed: single, double and treble, there are
exactly 82407 different ways to end the game! From 58 points you can finish in
1404 different ways! So, if you try to calculate the exact probabilities by
hand, you will probably lose a good hand for playing darts.

To be a bit more exact, I chose to use three different aiming points in each
bed: one in the middle, and two on both sides of the bed. As there still are 21
points in the bull, the computer uses 201 different aiming points when searching
for the optimal play.

Different aiming points are marked a bit differently in the book than you may
have seen before. For example 'T16(8)' means the point in treble-16 but closer
to sector 8. Marking the aiming points like this also helps you to realise to
which sector it is better to miss your dart - if you should miss it.

With the computer we can check every possible way to finish the game. Since
the calculating model is quite simple, the programmes are also easy to make. A
simple carefully made programme guarantees correct results. All you have to do
is to make the computer try every possible aiming point on the dart-board, and
search all possible - and sensible - areas the dart might hit. The computer must
then try to finish the game from each different point. If it's not programmed to
be selective, it has to again try every possible aiming point available. Note
that a beginner can very easily hit one side of the board when trying for the
other, but this shouldn't affect his choices.

The computer now plays with 201 different aiming points. Then it considers at
least 15 different places (treble, double and single - in five sectors) around
the aiming points where the dart may land. This makes already 3015 different
variations, but for example around the bull's-eye there are 22 beds which the
dart may hit. So I now simplify things a bit.

From every possible point it tries to continue the game again - with 201
different ways - and so on. For three darts the computer must thus check at
least 27.407.000.000 (3015^3) different variations of play. And after this it
has solved an optimised way to finish from only one point. In the game of 501
there are 162 different points where the player can finish the game. This means
calculating the probabilities for 4.440.000.000.000 times!

All this requires powerful computers to get results in a reasonable time.
Note that in the book I decided to go a bit further. Like in chess the computer
thinks ahead five turns, i.e. 15 darts! And the variations above were calculated
for three darts only. There were also more points than 162 to check this time.
The computer searched for the best play from every point under 501! This of
course means that you need to make a bit more complicated programmes to solve
the problems.

Opponent's situation

There is more. As you have noticed solving the probability of finishing is
only one part of the game. But there are always two players in darts and the
opponent can finish the game before you. Winning the game means that you must
finish first. To find the best way to play the computer must also consider the
opponent's moves. When figuring out the optimal finishes, the computer checks
the best finishes for every possible situation where the opponent can be. Note
that it's assumes that both players play the game with similar skills.

Since the subject is quite interesting, let me show you the way the computer
does the calculating. Let's take an example where a professional player has 90
points and three darts in hand. The opponent has 123 points. By playing T20-D15
a pro would still get to the bull with singles (20-20-B50). That is why T20-D15
is easier (47.8%) to finish with three darts than for example T18-D18 (46.5%):

An opponent of the same level is at 123 points and finishes the game in the
next turn with the probability of 24.3%. This means that the player can still
have a chance to win the game on the second turn with the probability of 75.7%
(100%-24.3%). This means that the player will win the game (on the next two
turns) with the probability of 78.5% (0.478+0.757·0.405) when trying the path
T20-D15. Because the probability for the path T18-D18 is better, 79.7%
(0.465+0.757·0.438), in this situation it's better to aim at treble-18.

Simple, isn't it!

Searching for the optimal play in 501

Finishing the game of 501 (301) is the most interesting part of the game. As
in every sport you can learn a lot by watching how the masters play.

But the problem is that all players have their favourite doubles and finishes
from each point. On the other hand, the level of play may be quite different
from yours. There is no sense in trying to end the game when it would require a
miracle and leads to a worse situation than in normal play.

The difference between several great endings is often small. This gives room
for variations of different throws and only the worst ones can be claimed to be
wrong. As a conclusion you may say that it's impossible for anyone to give exact
rules for how you should finish your game.

It would, however, be interesting to know the exact and best way to play the
game. This is not only for memorising - which of course gives you a big
advantage - but analysing the game is also a very useful way to improve your own
game.

Computer throws darts

This was the reason to make a computer throw darts. You can programme
different dispersions of darts to a computer and thus find the very best
finishes for each level of players.

Since the best throw means the best aiming points to win this game -
and not finishing on the next turn - you can also see when a player should go
for it and when to play it safe. With a more complicated programme you can also
include the opponent's situation in the game and find the best throw for every
'real' situation of the game.

There are only a few differences from the real world. For example, due to a
lot of practice you may find triple-20 a bit easier target than triple-19. If
the difference in probabilities in table is very small, you should choose T20 to
play. Better players, like professionals, should be able to hit any part of the
board with almost the same success. Since the computer has never 'practiced'
darts, it recommends triple-19, even when it is only a little better throw than
e.g. triple-20.

At first for a beginner and an average player triple-19 is the best choice
for scoring, but it is assumed that this will pass with a lot of practice.
Therefore the theoretical notation of the computer - that at bigger points a
beginner should use T19 to get points - is removed from the table.

How to read the table of optimised play

In the following table you will see some results of optimised play in 501.
(This table and lots of others are enclosed into The Dart Book. There you can
find the optimised play for each point - up to 501pts - also for the second and
the last dart in your hand.)

The best paths have been selected by a computer for four different levels of
players. Since the area of hits is very large for a beginner, the optimised
aiming point might sometimes be in a peculiar place. If for example the sectors
17 and 19 are the right areas to play, the computer may recommend aiming at
triple-3. This is no doubt the best place to aim at if the dispersion of darts
is big, but since this is in controversy with the basics of darts, I have
usually removed these paths from the table. In these cases it's better to
practice your throwing - and not start throwing into awkward places on the
board. There is a lot of other information hidden in the table.

For every level an exact dispersion is chosen so that exact percentages can
be calculated. So please remember that the name of the level only describes the
level of players. It doesn't mean that e.g. every 'pro' scores 180 points every
tenth time. It only makes the table more understandable.

As mentioned in the book getting to small points doesn't always mean that you
have better possibilities to finish the game. The local tops of the winning
probabilities (when the opponent has those 32 points) are marked with exlamation
marks (in the book it's a small dot).

For maximum accuracy the computer may recommend that you don't use the middle
of the bed but one side of it. "T19(7)" means that the best aiming
point is in triple-19 but closer to sector 7. With the help of the computer it
is possible to find an even more exact point, but it would be more difficult to
mark it.

Marking the finishes this way also helps the player to understand the game.
You can see that if you miss sector 19, it will be better to avoid sector three
and hit sector seven. Moving the aiming point, however, means that you don't hit
the best target as often. Since the exact side point is programmed to be 6°
(from the centre) to the other side in every bed, the computer may recommend
another side for the average player and the centre of the bed for experts. This
doesn't mean that the optimal target for an expert is in the centre - but closer
to it than to the side point. So, if you read carefully, you also receive
information between the lines.

The order of the rows is not random. For each level of player the first path
is the best when the exp-level opponent has 32 points (D16). This means the most
critical - and perhaps most important - situation. The second path is better
than the third one and so on.

Note that there are some points where the outer bull (25p) is the best way to
continue the game. But, as proved in the book, the best chance for all levels to
hit the outer bull is when you aim at the centre of the bull. (This is when the
inner bull doesn't lead you to a bad situation, like busting.) Only if you are
better than the 'professional' mentioned in the book, do you start to get an
advantage from moving your aiming point to the outer bull. Therefore, if 25
points is the best way to continue the game, the best aiming point is given in
parenthesis "B50(25)".

The probabilities of finishing are also listed. There you can find the
finishing percentages for the turn (3 darts). The computer optimises your play
five turns ahead, so it "plays" the game 15 darts ahead - this means
it optimises the game from the very beginning of 501! There are no limits for
planning the game even further ahead, but the following turns have little impact
on this turn.

Note that the probability of finishing doesn't mean the probability of
exactly throwing the mentioned path! That is not important. The probability
means the chances of finishing the game when you start the turn by aiming at the
given point. All the possible mistakes and the different variations they cause
are taken into account in the percentage.

The limits show you at what points the opponent may be when the path is
recommended. The level of the opponent should now be the same as the player's -
as in most cases it is.

The table of optimised play with the first dart

The following table and lots of others are enclosed into The Dart Book. There
you can find the optimised play for each point (up to 501pts) - also for the
second and the last dart in your hand.

As you see some of these results are quite amazing. Please note that these
figures are the "mathematical facts" of the sport and there is nothing
I can do to change them - even if I wanted to! In most cases you can however
quite easily comprehend the computer's way of thinking and find out why the
specific path was recommended.